In this note we discuss a slight
generalization of the following result by Alt and Caffarelli: if
the logarithm of the Poisson kernel of a Reifenberg flat chord arc
domain is Hölder continuous, then the domain can be locally
represented as the area above the graph of a function whose
gradient is Hölder continuous. In this note we show that if the
Poisson kernel of an unbounded Reifenberg flat chord arc domain is
1 a.e. on the boundary then the domain is (modulo rotation and
translation) the upper half plane. This result plays a key role in
the study of regularity of the free boundary below the continuous
threshold.